We went here for our honeymoon many years ago. Unfortunately that meant December. It was cold, and dark and miserable. The trip didn’t end well, but that’s not Copenhagen’s fault (no, nothing that dramatic - still very happily married!). Go when there’s actual daylight, and it’s a nice spot, but it’s expensive if you don’t have a way to self-cater! The Danes have put that burning and pillaging thing far behind them, and were universally friendly and helpful - most people we spoke to were surprised and pleased that we’d decided to come to their country for a holiday.
Unless you're an influencer who needs the picture for your Instagram, skip the Little Mermaid. It's a long way away, not that big, not very close to shore, and that shore is always mobbed by coach parties.
The Danish National Museum (on Ny Vestergade which is not far from Tivoli, but is confusingly quite a distance from “old” Vestergade!) is worth a morning if you’re interested in early European history. In typical Scandinavian fashion, it’s laid out chronologically - you start in pre-history at the first hall on the ground floor, and end in modern times at the top. There’s too much for one trip, but the section up to 1000 AD is fascinating. The Design Museum isn’t bad either, but if you only have one “culture” thing, do the National Museum.
For shopping, the Illums Bolighus on Amagertorv is an Aladdin’s Cave of designer furniture and homewares. Actually buying anything here requires very deep pockets, but if you’ve ever shopped in Scandium on Marylebone High Street in London, you’ll find it almost economically priced.
Malmö is worth a day-trip and a walk around. Sadly, the train goes on the lower deck of the Øresund bridge, so you don’t get the spectacular view that car-drivers get; but then again, they have to pay £50 to enjoy it…
Everyone speaks English: the Danes expect nobody outside of Scandinavia to be able to speak their language, which they regard as impenetrable to foreigners (As a Danish colleague once said: “How do you learn Danish? Become fluent in Swedish, then stick a very hot potato in your mouth”). That said, a well-placed “tak” (thank-you, and also “please”) goes a long way.